Zeroes of Sine and Cosine
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Theorem
- $(1): \quad \forall n \in \Z: x = \paren {n + \dfrac 1 2} \pi \implies \cos x = 0$
- $(2): \quad \forall n \in \Z: x = n \pi \implies \sin x = 0$
Proof
From Sine and Cosine are Periodic on Reals: Corollary:
$\cos x$ is:
- strictly positive on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$
and:
- strictly negative on the interval $\openint {\dfrac \pi 2} {\dfrac {3 \pi} 2}$
$\sin x$ is:
- strictly positive on the interval $\openint 0 \pi$
and:
- strictly negative on the interval $\openint \pi {2 \pi}$
The result follows directly from Sine and Cosine are Periodic on Reals.
$\blacksquare$
